LEAST SQUARES ESTIMATION FOR A MULTIVARIATE WEIBULL
MODEL OF HOUGAARD BASED ON ACCELERATED LIFE TEST
OF COMPONENT AND SYSTEM
Jye-Chyi Lu
Department of Statistics
North Carolina State University
Raleigh, NC 27695
Key Words and Phrases:
Component and system data; Multivariate Weibull;
Least Squares Estimator; Asymptotic Distribution.
ABSTRACT
Accelerated life testing of products and material under severe conditions
quickly yields information on life. In this article, we present a simple method to
incorporate the information collected from the accelerated life test on component
and (series) system levels.
The underlying distribution of the lifetimes of the
components is assumed to be a multivariate Weibull due to Hougaard. The least
squares estimators of the model parameters are proposed along with the
derivation of their asymptotic distribution.
1. INTRODUCTION
Accelerated Life Test (ALT) is commonly employed when product or
material reliability is high and testing under normal use condition would make test
time prohibitively long.
With ALT, test units are subjected to stress conditions
that are more severe than those encountered in normal use so that more failures
are apt to take place in a limited time.
Data of failure times under such over-
stress conditions are drawn with regard to life length or reliability of the product
under its normal use condition.
applications.
Nelson (1974a) provided a bibliography of
Bhattacharyya (1986)
reviewed
the principal methodological
approaches to ALT analysis in regard to plausibility of the model, flexibility of
empirical fit and usefulness in practical application.
Before a system is developed, ALT on component levels are usually
conducted (see examples in Mann, Schafer and Singpurwalla, 1974).
When the
components are assembled into a system, life-test data may also be obtained from
system testing.
Nelson (1973, 1974b), Klein and Basu (1981a, 1981b, 1982),
analyzed the ALT data from series system testing.
When life test data are
available for a system as well as the components, it is desirable to utilize all
available data to improve the component reliability estimation and system design,
particularly in situations where the available data are limited.
With regular life
test on normal use condition, Miyamura (1982) analyzed life-test results of the
electromagnetic valve a.nd of the air conditioner, which is a series system of the
electromagnetic valve and other components.
Easterling and Prairie (1971) gave
an example on testing of parallel system: for a certain thermal battery with two
bridgewires, life test data may be obtained at the component level (bridgewires) as
well as the system level (battery).
With the assumption of the exponential
lifetimes, Easterling and Prairie and Miyamura used the method of maximum
likelihood, and Mastran (1975) presented a Bayesian procedure to estimate the
parameters of the distribution of component lifetimes. In the context of ALT, the
estimation
method
based on
component and system data have not been
adequately developed.
In all these aforementioned studies, the component lifetimes were assumed
to be independent for the sake of simplicity of mathematical treatment. In a twocomponent system, the failure rate of one component might be increased upon the
failure of the other component.
Common cause failure or similar environmental
factor (stress) might lead to the dependence of the components.
Modeling the
lifetimes of the components as Gumbel's (1960) bivariate exponential BVE, Lu
and Bhattacharyya (1988b) developed several exact inference procedures based on
the data obtained from the regular life test.
In this article, we consider a system of m identical components whose
lifetimes may be dependent due to the effect of common environmental factors.
We develop estimation methods for the life length of the system based on ALT
-2-
data from series system and component testing.
from the following consideration:
This problem is also motivated
. sometimes, due to the limitation of time or
cost, the experimental stress were set far from the normal use condition.
Consequently, an extrapolation to the use condition may be quite unreliable. To
narrow the gap between experimental and use condition or to check the model for
extrapolation, one would like to observe some data at stress closer to the usecondition. Since the failure time of a series system is the minimum of the failure
times of its components, testing of a series system might be conducted under
lower stress. Hence, besides analyzing the result from component testing, utilizing
the information from life test of a series system allows us to check acceleration
model in lower stress.
In Section 2 we introduce the set-up of the experiment. We apply a flexible
and physically motivated distribution to the lifetimes of both system and
components (under each stress).
Between different stress levels, a stress
acceleration function is formulated. In view of the computational complexity and
the lack of closed-form solution for maximum likelihood (ML) estimation, we
propose a set of two-stage least squares (LS) estimators in Section 3. Section 4 is
devoted to the derivation of the asymptotic distribution of the LS estimator. We
consider' both the cases of the number of replications tends to infinity and the
number of stress settings tends to infinity.
2. LIFE DISTRIBUTION OF COMPONENTS AND SYSTEM,
AND STRESS ACCELERATION FUNCTION
Let us denote the lifetimes of m components in a system as Zl' ...• Zm and
suppose that they have the same distribution individually, regardless if the
components are assembled into a system.
We apply the Hougaard (1986)
multivariate Weibull distribution MVW to model lifetimes of the components.
The survival function (SF) of the MVW is of the forF
F(zl' z2' .. ' , Zm)
6 E (0, 1], Pi' 0i
This
distribution
= exp [- k=l
E (Zk/ Ok/k/ 6]
> 0,
t
= 1, ...
enjoys several
, k,
Zl'
important
,
,
zm
? 0.
properties
(2.1)
such
as
a
physical
motivation (cf. Hougaard, 1986), existence of absolutely continuous probability
density function, Weibull marginals and minimum for equal shape parameters case
(or stability relation phased in Tawn, 1988). Its bivariate case can be obtained by
-3-
using a
power transformation of Gumbel's
BVE which
has been studied
extensively by Lu and Bhattacharyya (1988a, b) for paired and system-component
data collected from regular life testing procedures.
We consider the following set-up of the accelerated life test.
On the
component level, we suppose that an experiment is conducted under k 1 stress
levels £i'
i
= 1, ... , k i .
At
stress £i'
components are put on
ni
test
simultaneously and the experiment stops as soon as the first r i failures are
observed (type II censored sample). The lifetimes of these n i components are
modeled as independent random variables (rv) Zil' ... ,Zin,' The type II censored
I
sample refers to a specified subset of the order statistics Zi(l)
Zi17 ... ,Zin,'
Zi(ri) of
For system testing, we suppose that an experiment is conducted
I
=k -
under k 2
< ... <
k i stress levels £i' i
the type II censored sample Z'(I)
I
IJ
independently identical rv's ZiIIJ"'"
= k i + 1, ... ,k.
< ... <
At stress £i' we observe
Z,( ) ' which is a subset of the
I
ri
IJ
ZinilJ' where
ZijlJ'
j
= 1, ... , n i
are the
lifetimes of series systems.
Under a stress
XI"
.....
we assume that the random variables Z I).. , j
= 1, ...
, n·I
are independently identically distributed (iid) as Weibull distribution with the
sc~e
and shape parameters () and {3, respectively. Since the series system consists
of m identical components which have the MVW (2.1) as joint distribution, the
system life ZijlJ then has the Weibull distribution with the scale and shape
parameters m -6/IJ() and {3, respectively.
For the stress acceleration function between different stresses, we assume
that the parameter () depends on a p-vector stress according to a log-linear
relation log () i
£i
= £i 2, where
= (XiI'
... ,xip)',
2
= (0'1'
... ,O'p)',
J
= 1, ...
,
k,
while the dependence (6) and shape ({3) parameters are independent of stress.
The assumption of a log-linear relation to stress is not only simple and flexible but
is also motivated in many practical contexts. The Arrhenius reaction rate model,
Inverse power law and Eyring model are some of the widely used engineering
models which fit into the log-linear relation.
3. LEAST SQUARES ESTIMATION
The
method of maximum
likelihood estimation involves considerable
computational complexity and, lacks a closed-form solution.
-4-
An analytical
treatment of exact properties of the ML estimators does not appear to be feasible.
In the context of ALT, for the analysis of results from component testing, some
interesting procedures have been developed for the life distributions in locationscale family. A simple estimation procedure with type II censored data, proposed
by Nelson and Hahn (1972, 1973), is based on an application of the least squares
method in two stages. This method leads to unbiased estimators of the mean and
any percentile log-life as well as their exact variances as opposed to only
asymptotic results obtainable for the MLE's.
For a given stress of component testing, the log-life Y (= log Z) is written
as
Y = log 0
+
11 W,
where W has the standard extreme-value distribution with probability density
function (pdf)
exp
[W -
exp (W)],
-
<W <
00
00.
Similarly, for a given stress of system testing, the log-life T(
= log Zs)
is written
as
T
= log 0 -
11 0 log m
+ 11
W.
Since these are linear regression models, least squares estimation based on order
statistics can be used.
To combine these two linear models together, we define
= 011, ,\ = log 0,
{ 0
for i = 1, ... , k 1 ,
(i = - log m
for i = k1 + 1, ... , k.
aj = E( Wi(j))'
Note that a/s and (/s are known constants while r, ,\ are unknown parameters,
r
,\ depends on £, by linear relation and
r does not depend on
£,. The observed log-
life from component and system testing can be presented by a general linear model
of the form
Yi(j)
= ,\
+
11 a j
+
r(i
+
eij'
j
= 1, ... ,ri'
i
= 1, ... , k,
(3.1)
where ei/s have mean 0 and covariance matrix
2
11 (U j j')
= 11
2
Cov [Wi(j)' W iri, )}
The means and covariances of Wi(j) are known constants (tables available, e.g.,
White, 1964).
The linear unbiased least squares estimation is obtained in two stages. In
the first stage, we ignore the regression structure and estimate the parameters
-5-
(Ai' "1i' 'Yi) from the ith data set, Yi(1)
$ ... $ Yi(ri) through the least square
estimation
= k1
method.
However, for
estimable parameters are Tj = Aj
i
+ 'Y(j,
+
1, ... ,k (system testing)
the
and "1 due to the deficiency in rank.
Hence, we should rewrite the linear model (3.1) as
Let us denote
ri
<l Ii
ri
= .L: L:
..,
(T)),
)=1 /=1
r,
r·
•
1
L: L:
<l3i =
••f
G:jG:/{T)) ,
j=I/=1
where {Tj/ is the (j, j') element of the inverse of the covariance matrix ({Tjj')' We
thus have the stage-1 ordered linear unbiased estimators (O-BLUE) (cf. Lloyd,
1952) of the form
T*i
ri
=
L: a"y,(.,), "1t
j= 1
I)
I)
ri
=
E b .. Y'(.')'
j= 1
I)
I)
i = 1, .. , , k,
(3.2)
as well as their exact covariance matrix
"1
2i
2[ dJj
d2i
d ]
d3i '
(3.3).
where
_ <l3;
•
d Ij - <l,'
I
d
- <l 11'
3j -
<l.'
d
I
Hence, from the known values of
-
2i G:)'
-
<l2'I
--X:-.
I
and
(T ""
))
as well as
the coefficients a I).. and b..
I)
d1i , d2i and d 3i can be evaluated (e.g., White, 1964).
Remark:
simplify
Tt
In system testing if 6 = 1, i.e., independence case, we can
to:
Tt = At + "1t (- log m)
Then we can obtain the LS estimators of
At as follows:
-6-
r,
rj
I
A·I
.. log mJ
="
L.J [a IJ.. + bIJ
j=l
C ..
= ~,11 / E
r(~3j
=l L
y '(. 'J
I J
= j=
E1
=
+
C',
Y I'(.J'J' i
a"
(~lj - ~2j log mfl qJJ.
IJ
k1
1, ... , k,
where
r·I
IJ
In
A
the
second
= log () = X 2,
~2j log m)
-
stage,
we
+
take
,~
J
account
\*
\*
\*)' 2 * = (*
= (""l,"',""k'
TJ1,"·, TJ *)'
k.'
OJ
= diag
(djl' ... , djiJ,
Tj
of the
regression
structure
Let us define
~
We recall that
..J
j
= 1, 2, 3.
= Aj + 1(j = £' 2 + (j 1.
Using the estimators obtained in
the first stage, we form linear models of .....
T * and TJ * separately as
.....
T*
I"'W
= X* a* + #"'oWl'
e
I"'toJ
TJ* = .....
1 TJ
.....
+ .....e 2'
(3.4)
where X* = (X, .f)' 2* = (2', 1)', and their pair (~1' ..t2) has mean (Q, Q), is
independent across rows and has the covariance structure (3.3) across columns.
Based on these linear models, the weighted least squares unbiased estimator'
(WLUE) are obtained as
a*
.....
= (X*' 0-1 1 X*)-1X*' 0-1 1 .....T* '
7f = (!.' 0 31 ,V- 1 l'
0:1
1
k
2* = ,E d?Jl
k
TJt /
E d?Jl-
i= 1
1= 1
Since X* and .....
a* matrices can be decomposed into two parts, we further
separate the WLUE's, ..... and
as. follows:
r,
a
X' 0-1 1 X
('
.....
0- 1
1
X
X' 011 f ]_1[ X' 011 r..* ] .
(' 0-1 1 ~
(
('
~
I"'W
-7-
0-1 1 ~
T*
(3.5)
Let
A1
P
= X' D 11 X, A 2 = ('
""
1
= (X' D 1 X)-lX ' D 11,
D 11
Q
A3
(,
""
=I -
= X'
X P,
D 11
u*
(,
""
= ""('
D 11 Q ( .
""
Inverting the matrix in (3.5), we get
-lJ [X'D-1
A -1A
1
3 u*
-
U- 1
('
*
These yield the WLUE of 2
1
,.,.
r
,.,.
*]
D-1 r*
1,.,.
.
and r as
(3.6)
and their variances and covariances are given by
Cov(a)
,.,.
=
Var('1)
= 7]2 [,.,.( ' D-1
( 1,.,.
7]2
«('
[X' D-1
(
D-1
0- 1 ,.,.(' D-1
1 X - X' D-1
1 ,.,.,.,.
1,.,.
1 XJ-1,
(' D-1
(]-1,
,.,.
1 X(X' D-1
1 X)-l X' D-1
1 ,.,.
Based on these results, the simple estimators of shape and dependence parameters
can be constructed as
71 = 1(ii, '8
='1(ii.
Instead of formulating the linear models (3.4) of L* and ,.,.
7]*
separately, we can put them together into a single linear model and then apply
Remark:
the least squares method to obtain the best linear unbiased estimators (BLUE) in
one stage (cf. Lloyd, 1952; Nelson and Hahn, 1973).
However, this procedure
involves inverses of larger matrices than the ones in two-stage least squares
approach.
Moreover, the WLUE's are widely advocated in engineering; they are
highly efficient with respect to the BLUE's; they also provide information for
checking the correctness of the model (cf. Escobar, 1986).
a
Next, we discuss some special cases of ,.,. and
'1.
To simplify the notations,
we will drop all the subscripts of summations. For instance,
-8-
E
d-1 x r *
k
d-.. 1 x· r·*
=E
. 1 IJ
J J
~
L..3
X
2
k
=
~
L..
d-ij 1
2
Xj.
In a simple linear regression case, with p = 2 and
the weighted WLUE's
=
d-1
1
j=k 1+1
J=
Example 1:
ao
and
2
and
'1
can be simplified as follows:
6- {(E d11 r*)[E d11 x(x + l)J - (E d11 x r*)
1
x
[E d11 (x + l)J},
'ii, = (log
m)2
t:. -2 u;' {( 6,
- 62 )
[t:. '2:. r* - L: ar*]}.
""
"1 = - 6log m ( 6 E 3 r * - Ear *),
u*
where
u*
= (log m)2 1:::.- 1 {E3 d11 - (E3 d11)2 (E d11 x 2 )
+ 2(E3 d11) (E3 d11 x)(E d11 x) - (E d11) (E3 d11 X2 )2},
ai
= d1l
{(E
- (Xi
E3
d11
x2 )(E3 d11) + xi(E d11)(E3 d11 x)
d11 +
E3
d11 X)},
...... 9-
Example 2:
Following Example 1, if the replications and censorings are
equal in system or component testing, that is ni = ne, ri = re for i = 1,
+
and ni = n., ri = r. for i = k1
and dij = d 1 •
for i = k 1
+
1, ... , k, we have dij = dIe for i = 1,
1, ... k.
bj , j = 1, 2
ai and
6.,
Hence,
, k,
, k1 ,
in
Example 1 can be furthur simplified as follows:
6.
= (k 1 dI 1 + k2
- (d 11
ait
dID (d11
Ee x +
dI l
Ee
E"
x
2
+
E"
dI l
x
2
)
x)2,
= dil (di1 Ee x2 + dI l E" x2 ) (k2 diD + dil(xi
x (k 1 di1
+
k2 diD - dil [dil (k 2 Xi
where t = c for i = 1, ... , k 1 , t = s for i = k 1
+
+ E"
dil
E"
x)
x)J '
1, ... ,k. And,
If the replications and censorings are all equal at all stress
Example 3:
levels, i.e., ni = n, r i = r, then d li = d1 for i = 1, ... , k and the weighted
WLUE's become unweighted WLUE's. That is,
r;
2* = (X*' X*)-l x*' 1:.*,
=
i
k
.E
77:,
1=1
we can decompose the
X*
and
2*
matrices to get the linear unbiased
estimators of 2 and 'Y as
'ii'=P(I-( (1;l('Q)r*,
f!"V
'1=(1;l('Qr*,
"...".,,....,,""""
,....,,""""
with variances and covariances
2
COV(2) = 77 [X' X - X'
Var('1) = 77
£ ($,;' £)-1 £' X]
-1
'
2[£' £ - £' X (X' X)-l X' £J-1,
COV(2, '1) = - (X' X)-l X' ,...
( Var('1) ,
-10-
where
= (X' X)-l X',
p
In linear regression case, we have
20 = 6. -1{ (E X2 ) (E r*) + (E x) (E
.
- k
21
= (log m)2 6.- 2
fT;l
{(b 1
6.
= kE x 2 -
b1
= k2 E
x2 -
(Ex)2,
ai
(E"
(E
x)
= (log m)2 6.- 1 {k 2 -
E"
b2 ) (6.
-
= - (log m) 6.- 1 fT;l (6. E" r*
*
(E
x)
(E
x r*)
Ex r*},
1
fT
r*) -
E
-
r* -
E
a r*)},
a r*),
= (k 2 E x 2 ) + kXi E" x x),
k/
= k Ec x -
b2
(E
x
2
)
+2
k2
E
(k 2 xi
+ E"
x),
x,
k 2 (E" x)(E x) - k
(E"
x
2
)2 }.
4. ASYMPTOTIC DISTRIBUTION OF
THE WEIGHTED LEAST SQUARES UNBIASED ESTIMATOR
(0 7J, 6) is a function of the WL UE
,
e = (a, Ti, 1) , we only need to derive the asymptotic distribution of ......e. In the
Since the simple estimator
u,
......
......
......
......
study of the asymptotic distribution of WLUE, we consider two cases:
number of replications N =
settings k tends to infinity.
(i) the
Ie
E
ni
tends to infinity and, (ii) the number of stress'
i=l
In the first case, the number of stress settings k is fixed. We assume that
nJ N
e
-
11' i'
i
= 1, '" , k as N -
00.
Since ...... is a function of ......
r * and 7J *, we first obtain the asymptotic distribution of
......
C!:,*, 2.*).
In stage-I, the O-BLUE
and 7Jr are asymptotically efficient
rr
estimators of r i and 7Ji' respectively (cf. Bennett, 1952; Chernoff et al., 1967).
Hence, the joint asymptotic distribution of
.JTii [err -
ri)'
(7Jr -
rr and 7Jr is as follows:
7Ji)] ~ N 2 (Q, Eo),
z = 1, ... , k,
where Eo is the Cramer-Rao lower bound of the form
-11-
~
_
~O -
71
C.
2
1t
[
-
(4.1)
C2 i
and Cji' J = 1, 2, 3 are tabulated in Bain (1978). In view of the independence of
the observations in different stress levels, we obtain the asymptotic distribution of
- T
* and 71 * as follows
We introduce the following notations to relate the WLUE
Cj
= diag(cji' ... ,Cj/e)'
Qc
=I -
X Pc,
(T c
-
= (T -1
c..
1"
C-1
1
to (~1' .!:.2)·
= (X' C l 1 X)-l x' Cl 1,
Pc
= (' C l
-
1
-
Qc (,
To simplify the expressions, let us denote
B2
I
Q c,
Since the elements of the covariance matrix (3.3) of O-BLUE
(r:, 71:>
converge to
the corresponding elements of the covariance matrix (4.1), we have Dj
Cj
1 _
1,
j = 1, 2, 3. Then, we have the following asymptotics:
-
-It''D-1Q
1
-
(T*..
B 2'
Q' D'2 1 . U- 1 l' D'2 1 - B 3, as N - 00.
Using the equality P(I - (
~
X* 2*
+ .!:.1
.[N
(T;1 ( '
in the expression of
(2 -
2) -
f"'t",I
2
B~ (.[N .!:.1)
-
D'2 1 Q) X a = a, we can replace r* by
"'"'J
~
(3.6) to conclude that
= 01' (1).
·Similarly, we have .[N ("1 - ,) - Bj(.[N .!:.1)
= 01' (1) and
.[N (ry -71) - Bj(.[N .!:.2) = 01' (1).
Since the joint asymptotic distribution .[N (.!:.1' ..t2) IS normal with mean
-12-
Q and
covariance matrix E we thus establish the asymptotic distribution of the WLUE
-e.
Theorem 1: The asymptotic distribution of
-IN[(2 -
2), (Tf - 11), (;:; - r)]
is normal with mean Q and covariance matrix E a , where
B~El
B1
=
Ea
[ Symmetric
In the case of the number of stress setting k tends to infinity, we establish
e=
=
the asymptotic distribution of the WL UE
lemmas. Let a)~ = (a).
-
and denote A
j
=
,
a· k)' and b
l' ... ,
= (.!!1' ...
1, ... ,p, and T p + 1
),
,.!!p).
k
=.E
1=1
-
For k
bi Y i .
-
(li', Tf, '1) by using a series of
(b 1 , ... , bk )' he vectors of constants
= 1, 2, ...
k
let us define T).·
=i=1
E
a). i Xi'
'
We need the following assumptions for a
general result.
ASSUMPTIONS:
-
-
AI. The sequences a)~, j = 1, ... , p and b' satisfy
t
(i)
a)? i
i= l '
( ii)
t
= 1,
i=1
max a? -- 0,
i~k
= 1,
bf
k
= 1, 2, ...
max b? -- 0,
i~k
),1
as k --
I
00.
A2. The linear functions T , j = 1, ... , p, are orthogonal, that is for all j
i
a~ a.,
and k,
-) -)
-
A3. a~ D b =
-)
D
l
= O.
k
E
i= 1
=I-
a·· b· u· -- d). as k -),1
I
00,
I
j
= 1, ...
, p, where
= diag (Ul'"'' Uk)' and ui is the variance between Xi
and Y i "
An exercise of Lindeberg's Theorem gives the following lemma which is a
modification of Lemma 1 of Bhattacharyya and Soejoeti (1981).
Q
Lemma 1: Let (Xi' Vi)' i = 1, 2, ... be independent random vectors with
12 ) = 0 as k -- 00,
mean, unit variance, covariance U i' and lime B k /
r/
= (El
=
where B k
qir l3 and qi
EI X d 3 (and EI Yd 3 ) exist for each i, and ui
is a known constant. If Assumptions A1- A3 are satisfied, then
-13-
T = (T 1 ,
where
ET
...
,T p , T p + 1 )' ~ N p + 1 (Q, E T
= [~, ;]. i£ = (dJ , ••• , d
Remark:
),
p )'.
Because that the numbers of replications in any two stress levels
might be different, the distributions of
Ti
and
77i,
i
= 1, ... , k (as well as
e2 i) given in (3.2) generally are not the same.
Liapunov's condition on the moments of
Xi
eli
and
Hence, we need to impose
and Yi (or
eli
and
e 2i
in next
lemma). This assumption holds for the extreme-value distribution.
Let S 1
= X'
D I 1 X, and C 1 be the symmetric matrix such that C~
= S 1.
Similarly, we also define
Di 1/ 2 = diag(di// 2 ,... , dik1/ 2 ), D/l2= diag(di
y2,..., d/,(2), i = 1,2,3.
-=-
We denote Q k
Q* _k -
Tk1
(Q*, tk' sk)', where
C-1 X' D-1 e
1
1 _1'
tk
= 1ki l "~'
sk -
D-1
1 £1'
1k1 -I'D-1
2 £2·
The assumptions of the asymptotics of design matrices are as follows:
ASSUMPTIONS:
Bl. The limit of k- 1 S 1 exists and is a nonsingular matrix denoted by B.
Similarly, we assume the existence of k- 1 S2 where both
w1
and
w2
w1' k- 1 S3 -. w2 as k -
are constants.
-
00,'
B2. The limit of k- 1X' D'i'1 ( exists and is denoted by c. We also have
,..,
k- 1X' D'i'1 D 3 D 21 1:. - !! and k- 1 1:.' D 21 D 3 D'i'1.£ - s, as k - 00.
Lemma 2: If Assumptions Bl and B2 are satisfied, then
E1
=
-
B- 1/ 2 c w1-1/2
I pxp
(B- 1/ 2 £, wl 1/ 2 )'
1
(n- 1/ 2
where the limits
n, £" !!'
!!
-1/2
w;1/2)' w 1
-1/2
w2
S
n- 1/ 2
-1/2
w1
-
d w2 -1/2
-1/2
w2
S
1
w 1, w2' and s are defined in Assumptions Bl and B2.
-14-
To prove this lemma, we shall make use of Lemma 5.1. We
Proof.
consider the independent random vectors (Zli' Z2i' Z3i)' i = 1, 2, ... , where
Zli
= Z2i = a;//2 eli
and Z3i
= a;/l2 e 2 i'
Thus (Zli' Z2i' Z3i) has mean
and unit variance. The covariance of (Zli' Z2i) and (Zti' Z3i)' t
= d2J(d1id3i)1/2.
d;i' respectively, where d;i
A
= D 11/ 2 X
b'2-_
= (2.1' ... , 2p),
1
C1
Q
= 1, 2, are 1 and
Let us define
D-1 1/ 2 _( C-1
2'
b' -_1
1 2
D2 / _1 C-1
3'
where 2j denote the jth column of the matrix A.
Then,...[k
...[k sk can be written in terms of ~1' ~2 and Z3 as ...[k
and ...[k sk = !j~3' respectively.
9: =
9:
and ...[k t k and
A' X, ...[k t k = !~ Z2'
Next, we show that under Assumptions Bl and B2, the Assumption AI-A3
hold.
By Assumption Bl A'A
conditions A l(i) and A2 hold.
!~!2 -
s as k -
00,
i~k
1
'e
'
p
< 1::
-
= Lp, !j!2 = 1,
A!j - i
A!~ - £.
the
and
and.1:: xJ'i
1=1
every j
for
Ik s''e I max {Xji Xei } .
p
1::
i~k
j=l e=l
Xji xe/(dij k) -
TJ.1ax aJ~ i :.... 0
I~k
By Assumption B2
p
By Assumption Bl, k SJ
have
D"j1 X 01
so the Assumption A3 holds. Denoting the ith diagonal
2 X S-l X' D- 1/ 2 by q we have
1
1
i'
max I q.,
'rll
x'
D-1 1/
ele.ment of AA' --
implies that
= Cj1
0,
d 1i k
e1./(d1i k) are convergent, andP the latter
so 'r<allqil - O. Since qi = j~l aJ,i' we·
X
= 1, ... ; p.
This
result
along with
the
convergences of S2/k and S3/k lead to the Assumption Al (ii). Finally, a direct
-
appliation of Lemma 1 leads to the desired asymptotic distribution of ...[k Qk'
0
Let us define tA; = ~ C 2 t k and sA; = ~ C 3 sk' The following lemma gives
~k
~k
the asymptotic distribution concerning (P, tA;, sA;).
Lemma 3: Let P be equal to (X' D1 1 X)X' D 1 1 and tA;
sA; -
t
...[k
C 3 sk' If Assumptions Bl and B2 al'e satisfied, then
[p ~1' tA;, sA;]
~
N p + 2(Q, 1:: 2 ),
-15-
=t
C 2t k and
where
E,
~
[
B-1
B-l£
B-1 fl
(B-1 £)'
wI
S
(B-1 fl)'
S
W2
ci = SI = X' D l X
Proof. By
Therefore, from Lemma 2 ..[k
Q and
mean
[h c
J
we have
1 P f:.l' t
k, SkJ
is asymptotically normal with
covariance matrix E I" By an application of Delta method along with
1 .
the asymptotic 1":' C 1 -
B
~k
1/2
o
, we complete the proof.
.
Lemma 4: If Assumptions Bl and B2 are satisfied, then
where
E3
=
B-1
0
'"
B- l / 2 d
'"
0'
'"
(7a
(7b
(7b
w2
l
(B- / 2
an d
(7
a
= wI
-
£ ' B-1 £. an d
Proof. By Q
t f'
flY
D1 1 Q
=I
f:.l
(7 b
=S-
£
I
B-1 '"
d "
- X P we have
= ..[k tk -
(l f' D11 X) ( ..[k P f:.l).
From Assumption B2, we know that
l f' D11 X -
£' as k
-00.
Hence, the
asymptotic distribution of ~ (' D 11 Q e 1 is the same as the asymptotic
~k '"
'"
distribution of ..[k t k - £' (..[k P f:.l).
Using the joint asymptotic distribution of
..[k (P f:.l' t1;, s1;) in Lemma 3, we complete the proof.
0
The following asymptotics are readily to be obtained by Assumptions Bl "
and
B2.
(i)
-16-
(7 * = (' 0Il Q (.
....
The asymptotic distribution of the WL UE
....
I.- is given in the
following theorem.
Theorem 2:
If Assumptions Bl and B2 are satisfied, then the joint
asymptotic distribution
Q and
of.Jk
[(2 - 2)' C::;
- ,), (17 -
7])J is normal with mean
covariance matrixE, where
E=
Ep
E p ,.
E p '1
E~,.
(7-1
a
-1 (7b
(7a
E~'1
(7a -1 (7b
w-2 1
and
E p =B-1[I+c(w
-c'n- 1 ........
c)-l c 'B-1] '
....
1....
~
-
....p,. -
B-1 ..£ ( wI
-
-
..£ , B-1 ..£ ) '
(7b
=
't"'
4JP'1
....c'
S -
[d
= B-1 .... - ..£ (7 a -1 (7bJ-1
z ,
B-1 ....
d.
Proof: We first recall that
.Jk (2 - 2)
=
.Jk P [I
-
f
(7;1
.Jk Cr - ,) = .Jk (7;1 ....(' 0Il
{k
As k -
(17 -
00
Q
f' 0Il Ql~l'
e
.... 1
,
1 1)-11' 0- 1 e .
7]) = {k (1'
0-2....
~
....
2 .... 2
we have
.Jk (2 - 2) - (.Jk
.Jk ('1'
P
~1) -
7]) -
w"2
1
0Il Q
1
in
Lemma 4,
the
0Il Q
~1) =
op (1),
= op (1),
)
Since the joint asymptotic normality of {k
established
(h f'
(t £' ~1)
.Jk (t l' °"2 ~2 = .Jk (17 -
- ,) - (7ri 1 .Jk
.Jk (17 -
(B-1 ..£ (7;1)
w2
.Jk Sk = op (1).
(p ~1' t f' 0Il Q ~1' Sk)
joint
-17-
7]) -
asymptotic
has been
distribution
of
..[k
[(2 - 2), Cr -. I),
Remark:
(1'f - 7J)J is readily established as stated in theorem.
0
If the dependence parameter 8 is in the interior of (0, 1], the
asymptotic distribution of
(11 u, 73,6)
follows from Delta method. For the case of
8 equal to 1, the correct limit distribution is obtained by an application of Eq.
(2.2) given in Self and Liang (1987).
BIBLIOGRAPHY
BAIN, L. J. (1978).
Statistical A nalysis of Reliability and Life- Testing Models,
New York: Dekker.
BENNETT, C. A. (1952).
'Asymptotic Properties of Ideal Linear Estimators,'
Ph.D. dissertation, University of Michigan.
BHATTACHARYYA, G. K. (1982).
Accelerated Life Tests: An Overview and
Some Recent Advances. Technical Report No. 783, University of Wisconsin,
Madison.
BHATTACHARYYA, G. K. and SOEJOETI, Z. (1981). 'Asymptotic Normality
and Efficiency of Modified Least Squares Estimators in Some Accelerated
Life Test Models,' Sankhya, 43, 18-39.
CHERNOFF,
H., GASTWIRTH, J.
L., and JOHNS, M. V., Jr. (1967).
'Asymptotic Distribution of Linear Combinations of Order Statistics, with
Applications to Estimation,' Ann. Math. Statist., 38, 52-72.
EASTERLING, R. G., and PRAIRIE, R. R. (1971). 'Combining Component an(f
System Information,' Technometrics, 13, 271-280.
ESCOBAR, L. A. (1986).
'The Equivalence of Regression-Simple and Best-
Linear- Unbiased Estimators With Type II Censored Data From a Location
Scale Distribution,' Journal of the American Statistical Association, 81, 210-
214.
-18-
GUMBEL, E. J. (1960).
'Bivariate Exponential Distribution,'
Journal of the
American Statistical Association, 55, 698-707.
HOUGAARD, P. (1986).
'A Class of Multivariate Failure Time Distributions,'
Biometrika, 73, 3, 671-678.
KLEIN, J. P. and BASU, A. P. (198Ia).
Accelerated Life Testing Under
Competing Exponential Failure Distributions.
Technical Report No. 108,
University of Missouri, Columbia.
KLEIN, J. P. and BASU, A. P. (1981b). 'Weibull Accelerated Life Tests When
there are Competing Causes of Failure,'
Communication in Statistics, AIO,
2073-2100.
KLEIN, J. P. and BASU, A. P. (1982). 'Accelerated Life Tests Under Competing
Weibull Causes of Failure,' Communication in Statistics, All, 2271-2286.
LLOYD, E. H. (1952).
'Least-Squares Estimation of Location and Scale
- Parameters Using Order Statistics,' Biometrika, 39, 88-95.
LU, J. C. and BHATTACHARYYA, G. K. (1988a).
Bivariate Exponential Model of Gumbel.
Inference Procedures for a
Technical Report No. 838,
University of Wisconsin, Madison.
LU, J. C. and BHATTACHARYYA, G. K. (1988b). Inference Procedures for a
Bivariate Exponential Model of Gumbel Based on Life Test of System and
Components.
Technical Report No. 855, University of Wisconsin, Madison.
MASTRAN, D. V. (1976). 'Incorporating Component and System Test Data Into
the Same Assessment: A Bayesian Approach,' Operational Research Society
Journal, 24, 491-499.
MIYAMURA, T. (1982).
'Estimating Component Failure Rates from Combined
Component and Systems Data:
Exponentially Distributed Component
Lifetimes,' Technometrics, 24, 313-318.
-19-
NELSON, W. B. and HAHN, G. J. (1972).
'Regression Analysis for Censored
Data - Part I: Simple Methods and Their Applications,' Technometrics, 14,
247-269.
NELSON, W. B. and HAHN, G. J. (1973).
Data - Part
II:
Best
Linear
'Regression Analysis for Censored
Unbiased
Estimation
and
Theory,'
Technometrics, 15, 133-150.
NELSON, W. B. (1973). Graphical Analysis of Accelerated Life Test Data with
Different Failure Modes. General Electric Company, Corporation Research
and Development Technical Information Series Report, 73-CRD-001.
NELSON, W. B. (19.74a).
Methods for Planning and Analyzing Accelerated
Tests. General Electric Company, Corporation Research and Development
Technical Information Series Report, 73-CRD-034.
NELSON, W. B. (1974b). Analysis of Accelerated Life Test Data with a Mix of
Failure Modes
by Maximum
Likelihood.
General
El~ctric
Company,
Corporation Research and Development Technical Information Series Report,
74-CRD-160.
SELF, S. G., and LIANG, K. Y. (1987).
likelihood
Estimators and
Likelihood
'Asymptotic Properties of Maximum
Ratio Tests
Under Nonstandard
Conditions,' Journal of the American Statistical Association, 82, 605-610.
TAWN, J. A.
(1988).
'Bivariate
Extreme
Value
Theory:
Models
and
Estimation,' Biometrika, 75, 397-415.
WAGNER, J. C., BERRY, G. and TIMBRELL, V. (1973). 'Mesotheliomata in
Rats After Inoculation with Asbestos and Other Materials,' British Journal
on Cancer, 28, 173-185.
WHITE, J. S. (1964).
'Least Squares Unbiased Censored Linear Estimation for
the Log-Weibull (Extreme Value) Distribution,' Journal of The Industrial
Mathematical Society, 14, 21-60.
-20-